(x%2B3)(x%2B5)%2B15.jpeg "(x^2+8x+7)(x+3)(x+5)+15")
Factoring and Simplifying the Expression (x^2+8x+7)(x+3)(x+5)+15
This article explores the process of factoring and simplifying the expression (x^2+8x+7)(x+3)(x+5)+15.
Step 1: Factor the Quadratic Expression
The first step is to factor the quadratic expression (x^2 + 8x + 7). We can factor this expression into:
(x^2 + 8x + 7) = (x + 1)(x + 7)
Step 2: Combine the Factored Expressions
Now we can rewrite the original expression:
(x^2+8x+7)(x+3)(x+5) + 15 = (x + 1)(x + 7)(x + 3)(x + 5) + 15
Step 3: Expand the Expression
To simplify, we need to expand the expression. We can do this by multiplying the factors together:
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First, multiply (x+1)(x+7): (x+1)(x+7) = x^2 + 8x + 7
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Then, multiply (x+3)(x+5): (x+3)(x+5) = x^2 + 8x + 15
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Finally, multiply the results from the previous steps and add 15: (x^2 + 8x + 7)(x^2 + 8x + 15) + 15 = x^4 + 16x^3 + 79x^2 + 128x + 112 + 15
Step 4: Simplify the Expression
Now we have:
x^4 + 16x^3 + 79x^2 + 128x + 127
Conclusion
Therefore, the simplified form of the expression (x^2+8x+7)(x+3)(x+5)+15 is x^4 + 16x^3 + 79x^2 + 128x + 127. This process demonstrates how factoring and expanding expressions can help us simplify complex mathematical expressions.